Say you have a complete Riemann manifold $R$ and a complete minimal hypersurface $M$ of dimension $n$ embedded in $R$. Say we remove all but but a patch $P$ from $M$ where $P$ is topologically $B^n$. Is $M$ the unique manifold that results from extending $P$?
It seems to me that in the case of $n = 1$, the statement is true. If you have a complete geodesic embedded in some complete Riemann manifold and you remove all but a small line segment of that geodesic, then extending that line segment in both directions gets you the original geodesic. I was wondering if this idea carries over to higher dimensions.